Correlation functions of CFTs on a torus with a T(T)over-bar deformation

In this paper, we investigate the correlation functions of the conformal field theory (CFI') with the T (T) over bar deformation on a torus in terms of the perturbative CFT approach, which is the extension of the previous investigations on correlation functions defined on a plane. We systematically obtain the first-order correction to the correlation functions of the CFI's with a T (T) over bar deformation in both operator formalism and path integral language. As a consistency check, we compute the deformed partition function, namely, the zero-point correlation function, up to the first order, which is consistent with results in the literature. Moreover, we obtain a new recursion relation for correlation functions with multiple T's and (T) over bar 's inserted in generic CFI's on a torus. Based on the recursion relations, we study some correlation functions of stress tensors up to the first order under T (T) over bar deformation.